Mazur's unpublished manuscript on primes and knots? - MathOverflow

Mazur's unpublished manuscript on primes and knots?

2010-12-13T19:32:12Z

The story of the analogy between knots and primes, which now has a literature, started with an unpublished note by Barry Mazur.

I'm not absolutely sure this is the one I mean, but in his paper, Analogies between group actions on 3-manifolds and number fields, Adam Sikora cites

B. Mazur, Remarks on the Alexander polynomial, unpublished notes.

He also cites the published paper

B. Mazur, Notes on étale topology of number fields, Ann. Sci. Ecole Norm. Sup. (4)6 (1973), 521-552.

I suppose an expert would recognize as relevance of this paper, but I don't see that even the word "knot" ever occurs there.

[My Question] Does anyone have a copy of Mazur's note that they would share, please? If not, has anyone at least actually seen it.

By the way, already years ago, I asked Mazur himself. I watched him kindly search his office, but he came up dry.

I realize that whatever insights the original note contains have doubtless been surpassed after c. 40 years by published results, but historical curiosity drives my desire to see the document that started the industry.

Answer by lieven lebruyn for Mazur's unpublished manuscript on primes and knots?

2010-12-13T21:28:24Z

$\newcommand\Z{\mathbf Z} \newcommand\F{\mathbf F} \newcommand\S{\mathbf S}$

Sadly, I don't have a copy of the unpublished note, but at least I can tell you what the published paper has to do with knots.

In that paper, Mazur shows that etale cohomology for Spec($\Z$) satisfies a sort of Poincare-duality as known for 3-dimensional manifolds. So, one might consider Spec($\Z$) as a 3-dimensional manifold. As its fundamental group is trivial, it should be after Poincare-Perelman the 3-sphere. Primes should be closed submanifolds in it and as the fundamental group of $\F_p=\Z/p\Z$ is $\hat\Z$, one might consider primes as circles embedded in $\S^3$, that is, as knots.

The analogy goes a lot deeper and I've written a blog-post about it some time ago (with references to Mazur's paper and follow-ups) :

Mazur's knotty dictionary

I've also given a talk about it, intended for a general public, last year. The slides are sadly in Dutch, but perhaps they convey what I was trying to tell : (34Mb download though ...)

What does a prime number look like?

Answer by David Feldman for Mazur's unpublished manuscript on primes and knots?

2011-02-15T04:12:50Z

This showed up in my snail-mail today, so I'm sharing the wealth:

http://ifile.it/rodc5is/mazur.pdf

Answer by Chandan Singh Dalawat for Mazur's unpublished manuscript on primes and knots?

2011-03-04T09:17:21Z

Thanks to this MO question, the paper Remarks on the Alexander Polynomial is now available on Barry Mazur's website.

Gratuitous addendum. For a recent overview, see

Morishita (Masanori), Analogies between prime numbers and knots, Sūgaku 58 (2006), no. 1, 40–63.

An English translation has appeared in

Sugaku Expositions 23 (2010), no. 1.